This collection comprises the expanded and fully refereed versions of
selected papers presented at the
31st
Conference on Computational Complexity (CCC 2016) held in Tokyo,
Japan, May 29 -- 31, 2016. These papers were selected by the program
committee from among the 34 papers that appeared in the
conference proceedings. Preliminary versions of the papers were
presented at the conference and the extended abstracts appeared in the
proceedings of the conference published by
Dagstuhl Publishing.
The CCC Program Committee selected
34 out of 91 submissions
for presentation at the conference; of these,
5 were invited to this Special Issue.
All papers were refereed in accordance with the
usual rigorous standards of
Theory of Computing.

Below are brief summaries of the five papers that appear in this
collection.

The paper “Average-Case Lower Bounds and Satisfiability
Algorithms for Small Threshold Circuits” by Ruiwen Chen, Rahul
Santhanam and Srikanth Srinivasan proves strong (inverse-polynomial)
correlation lower bounds for
bounded-depth
threshold circuits
computing parity and the generalized Andreev function, while
previously only
worst-case
lower bounds were known in these
settings. These results are then used to give satisfiability
algorithms for
bounded-depth
threshold circuits with a superlinear
number of wires which in turn imply
sublinear-exponential
($2^{o(n)}$)
learning algorithms for the related circuit class.

The paper “Proof Complexity Lower Bounds from Algebraic Circuit
Complexity” by Michael A. Forbes, Amir Shpilka, Iddo Tzameret and
Avi Wigderson studies proof complexity in the algebraic framework
using connections between algebraic circuit lower bounds and proof
complexity, a connection which has been well-explored in the Boolean
setting. In particular, this paper shows the first unconditional
lower bounds for versions of the Ideal Proof System, introduced by
Grochow and Pitassi (2014), restricted to various natural circuit
classes like depth 2, depth 3 powering circuits, read-once algebraic
branching programs, and various multilinear circuit classes, by
adapting techniques from circuit complexity.

Reed-Muller codes are one of the most
studied family of
codes. In particular, $m$-variate Reed Muller codes can be seen as
the evaluation of a low-degree polynomial on $S^m$ where $S$ is a
subset of the underlying finite field $\mathbb{F}$. While the
information-theoretic
unique decodability is not affected by the
specific choice of $S$ (and only depends on the size of $S$), the
algorithmic story used to be different. Previous algorithms either
required $S =\mathbb{F}$ or that the degree is considerably smaller
than $|S|$, the size of the set $S$. The paper “Decoding
Reed-Muller codes over product sets” by John Kim and Swastik
Kopparty fixes this gap in our understanding and gives an efficient
algorithm for unique decoding of RM codes for all settings where
the degree is at most the set size $|S|$.

The paper “Identity Testing for Constant-Width, and Any-Order,
Read-Once Oblivious Arithmetic Branching Programs” by Rohit Gurjar,
Arpita Korwar and Nitin Saxena provides a polynomial-size hitting
set for read-once algebraic branching programs of constant
width. Prior to this work only quasi-polynomial-size hitting sets
were known for this model.

The paper “Arithmetic circuits with locally low algebraic
rank” by Mrinal Kumar and Shubhangi Saraf demonstrates an explicit
exponential lower bound for depth-4 arithmetic circuits that are
sums of products of polynomials such that the set of polynomials in
each product is of low algebraic rank. Then, the authors construct
quasi-polynomial-size
hitting sets (and hence quasi-polynomial-time
deterministic PIT algorithms) for the same class of arithmetic
circuits with an additional restriction. These results give a clean
generalization of the recent results in algebraic circuit complexity
that dealt with depth-4 circuits.

I would like to thank the authors for their contributions and the
anonymous referees for their hard work that helped improve the quality
of this issue. It was a pleasure to edit this
special issue for Theory of Computing.